Question

The wavelengths for the light of red and blue colors are roughly $7.8 \times 10^{-7}\;m$ and $4.8 \times 10^{-7}\;m$ respectively. Which colour has the greater speed in a vacuum?
A). Red
B). Violet
C). Yellow
D). All colours have the same speed

Answer 1

Hint: Recall that blue and red lights are electromagnetic radiation in the visible region of the EM spectrum. Think about the speed with which all electromagnetic radiation travels. Also, we know that as you go across the spectrum, if the wavelength keeps increasing then the frequency keeps on decreasing. Use the relation between the speed, frequency, and wavelength to quantitatively arrive at your descriptive justification.

Formula Used:
The velocity with which an EM wave is propagated:
$v = \lambda f$, where $\lambda$ is the wavelength of light and f is the frequency of light.

Complete step-by-step solution:
We know that the light of red and blue colours is radiation from the visible region of the electromagnetic spectrum.
The electromagnetic spectrum is the range of all electromagnetic radiation known to us. These electromagnetic radiations travel through the vacuum of space in the form of waves. These waves are called transverse waves because the electric and magnetic fields constituting the electromagnetic radiation vibrate in a direction that is perpendicular to direction of propagation of the radiation.
Now, different electromagnetic rays are associated with different wavelengths and frequencies.
And the electromagnetic spectrum is classified as follows:

Radiation	Wavelength (m)	Frequency (Hz)
Gamma Rays	10-14 – 10-10	1022 - 1018
X-Rays	1 x 10-10 – 3 x 10-8	1018 - 1017
Ultra-violet (UV) Rays	3 x 10-8 – 4 x 10-7	1017 – 8 x 1014
Visible Light	4 x 10-7 – 8 x 10-7	8 x 1014 – 4 x 1014
Infra-Red (IR) Rays	8 x 10-7 – 3 x 10-5	4 x 1014 - 1013
Microwaves	10-3 – 0.3	1011 - 109
Radio waves	10 - 104	107 - 104

Therefore, if we go down the table, the wavelength increases, and frequency decreases for the radiation.
Now, let us look at our question.
We have blue and red colored lights whose approximate wavelengths are given.
In order to find the velocity with which they travel, they follow the relation:
$v = \lambda f$, where $\lambda$ is the wavelength of light and f is the frequency of light.
However, following the trend of the table shown, blue light has a lower wavelength but a higher frequency as compared to red, and red has a higher wavelength and a lower frequency when compared with blue.
Therefore, the product $v =\lambda f$ for both remains the same as the wavelength and frequency in each case numerically balance each other.
This constant value that we get is around $2.99 \times 10^8\; ms^{-1}$ which is nothing but what we call as the velocity of light.
This can also be numerically verified to an approximation:
(note that the following values taken are different from that specified in the question in order to get a good approximate)
Red light: $\lambda_{red} = 740\;nm$ and $f_{red} = 405\;THz$
$v_{red} = (740 \times 10^{-9})(405 \times 10^{12}) = 2.997 \times 10^8 \;ms^{-1}$
Blue light: $\lambda_{blue} = 485\;nm$ and $f_{blue} = 620\;THz$
$v_{blue} = (485 \times 10^{-9})(620\times 10^{12}) = 3.007 \times 10^8 \;ms^{-1}$, which is more or less equal to $v_{red}$.
Thus, we conclude that all electromagnetic radiation travels at the same speed in a vacuum.
The correct choice would be: D. All colours have the same speed.

Note: Remember that EM waves are transverse waves and do not require a material medium for their propagation and travel through free space. However, longitudinal waves, such as sound waves, require a material medium for the propagation of their compressional vibrations and cannot propagate through a vacuum. This is the reason why we cannot hear sounds in space and astronauts use radio waves instead to communicate with each other.